Integral of \( \displaystyle \cos^{2}{\left(3 x \right)} \)

Problem 4.223 · easy

1. \[ \int \cos^{2}{\left(3 x \right)}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \left(\frac{\cos{\left(6 x \right)}}{2} + \frac{1}{2}\right)\, dx \]
   trig-identityUse the power-reduction identity cos(2theta) = 2cos(2theta)^2 - 1, rewritten as cos(theta)^2 = (1 + cos(2theta))/2.✓ Proved
3. \[ = \int \frac{1}{2}\, dx + \int \frac{\cos{\left(6 x \right)}}{2}\, dx \]
   linearitySplit the integral into two separate terms.✓ Proved
4. \[ = \frac{x}{2} + \int \frac{\cos{\left(6 x \right)}}{2}\, dx \]
   antiderivativeIntegrate the constant term.✓ Proved
5. \[ = \frac{x}{2} + \frac{\sin{\left(6 x \right)}}{12} \]
   antiderivativeIntegrate the cosine term using the rule for cos(ax).✓ Proved